Exercise 1.4 (Basic properties of the simple integral)

Question

Let $f,g:\Omega	o[0,\infty]$ be simple unsigned functions.
\begin{enumerate}
	\item (Linearity) we have $\mathrm{Simp}\int_{\mathbb{R}^{d}} f + g = \mathrm{Simp}\int_{\mathbb{R}^{d}} f + \mathrm{Simp}\int_{\mathbb{R}^{d}} g$
	\item (Equivalence) if almost everywhere $f=g$, then $\mathrm{Simp}\int_{\mathbb{R}^{d}} f = \mathrm{Simp}\int_{\mathbb{R}^{d}} g$.
	\item (Finiteness) we have $\mathrm{Simp}\int_{\mathbb{R}^{d}} f < \infty$ if and only if (1) $f$ is finite almost everywhere and (2) its support has finite measure.
	\item (Monotonicity) if almost everywhere $f \leq g$ then $\mathrm{Simp}\int_{\mathbb{R}^{d}} f \leq \mathrm{Simp}\int_{\mathbb{R}^{d}} g$.
	\item (Markov inequality) Show that for any $0 < t < \infty$ we have $$\mu\left(\left\{\omega \in \Omega: f(\omega)\geq t ight\} ight) \leq \frac{1}{t}\mathrm{Simp}\int_{\Omega} f \,\mathrm{d}\mu$$
\end{enumerate}

Elu Thingol · Accepted Answer

We refer to the equivalent exercises from \href{..}{Terence Tao}'s \href{../an_introduction_to_measure_theory}{An Introduction to Measure Theory}:
\begin{enumerate}
	\item See \href{../an_introduction_to_measure_theory/exercise_1-4-33}{Exercise 1.4.33 (iii,iv)}.
	\item See \href{../an_introduction_to_measure_theory/exercise_1-4-33}{Exercise 1.4.33 (vi)}.
	\item See \href{../an_introduction_to_measure_theory/exercise_1-4-33}{Exercise 1.4.33 (vii)}.
	\item See \href{../an_introduction_to_measure_theory/exercise_1-4-33}{Exercise 1.4.33 (i)}.
	\item Let $f=c_1 1_{E_1} + \ldots + c_n 1_{E_n}$ be a simple representation of $f$ and assume that $c_1 \leq \ldots \leq c_n$. Let $1 \leq k \leq n$ be such that $c_1,\ldots,c_k < t$ and $c_{k+1},\ldots,c_n \geq t$. We then equivalently need to prove that
	$$\sum_{i=k+1}^{n} \mu(E_i) \leq \frac{1}{t} \sum_{i=1}^{n} c_i\mu(E_i)$$
	or, in other words,
	$$\sum_{i=1}^{n} c_i\mu(E_i) - \sum_{i=k+1}^{n} t\mu(E_i) \geq \sum_{i=k+1}^{n} (c_i - t_i) \cdot \mu(E_i) \geq 0.$$
	But this follows by the definition of $k$.
\end{enumerate}

Exercise 1.4 (Basic properties of the simple integral)

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Exercise 1.4 (Basic properties of the simple integral)

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